Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.
Given Two lines m and n are parallel and another two lines p and q are respectively perpendicular to m and n.
i.e., p ⊥ m, p ⊥ n, q ⊥ m, q ⊥ n
To prove p||g
Proof Since, m || n and p is perpendicular to m and n.
$\therefore$ $\angle 1=\angle 10=90^{\circ}$ [corresponding angles]
Similarly, $\angle 2=\angle 9=90^{\circ}$ [corresponding angles]
$[\because p \perp m$ and $p \perp n]$
$\therefore$$\angle 4=\angle 9=90^{\circ}$ and $\angle 3=\angle 10=90^{\circ}$ [alternate interior angles]
Similarly, if $m \| n$ and $g$ is perpendicular to $m$ and $n$.
Then, $\angle 7=90^{\circ}$ and $\angle 11=90^{\circ}$
Now, $\angle 3+\angle 7=90^{\circ}+90^{\circ}=180^{\circ}$
So, sum of two interior angles is supplementary.
We know that, if a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the
two lines are parallel.
Hence, p||g.
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